Sen. Patrick Leahy, D-Vt., chairman of the Senate Judiciary Committee, will play a key role in the debate over President Obama’s proposed gun controls.

WASHINGTON — As Senate Democrats grapple with how much of President Barack Obama’s broad gun control agenda is politically achievable, or even desirable, Sen. Patrick J. Leahy, D-Vt., chairman of the committee with jurisdiction over gun laws, said that moving to tighten background checks should be a top priority.

‘’One of the first things you want to do is close the gun show loophole,” said Leahy, in an interview for an appearance Sunday on C-SPAN’s “Newsmakers,” referring to a legal provision that permits unlicensed gun sellers to sell to people without conducting background checks.

The loophole is a small piece of a much broader agenda introduced by Obama on Wednesday, one that faces considerable, though not insurmountable, odds on Capitol Hill.

The view of Leahy, who has a history of supporting some gun rights, is crucial because the work of his Judiciary Committee will be central to advancing any new gun legislation.

He declined to detail precisely how the committee, which will hold hearings on potential gun legislation this month, would proceed with Obama’s request to push legislation that includes a renewal of an assault weapons ban, a limit on magazine size and universal background checks.

Leahy’s plans will also be closely watched by senators with their own gun control legislative agendas, lawmakers who may decide to push their own bills if the committee process does not yield the results they seek.

The 2004 expiration of the assault weapons ban tipped off the beginning of a new era in gun rights, one in which legislation began to tilt in favor of laws that strengthened, rather than chipped away at, gun owners’ rights to own all types of weapons, as well as where and how they could be carried.

An attempt to renew that law, which opponents in both parties say did little to stem gun violence in the 10 years it was in effect, faces formidable opposition.

Sen. Harry Reid of Nevada, the majority leader, who finds himself trapped between the desires of Obama and those of his conservative Democratic colleagues who would prefer not to have to vote on gun measures, has already signaled that that particular element of Obama’s agenda is in trouble.

‘’Is it something that can pass the Senate?” he said recently on a Nevada television show about a renewed assault weapons ban. “Maybe. Is it something that can pass the House? I doubt it.”

Leahy said, “I have told my staff I’d like to get a definitive answer if we can — ‘Did an assault weapon ban work?’”

It is a question that many Democrats and Republicans alike continue to pose.

It is possible that the Obama administration, which spent its first four years negotiating with Congress by offering its strongest hand up front, may use that legislation as a bargaining chip that it is willing to sacrifice to obtain the rest of its agenda.

Some Democrats, particularly Sen. Charles E. Schumer of New York, believe the best path to pursue is universal background checks for gun owners, which Schumer called “the sweet spot” among legislative paths. But such legislation would be scrutinized carefully for its details.

For example, in many states and rural areas, guns frequently change hands privately through sales to family members or neighbors, which could complicate background check procedures. But if written narrowly, a background check provision could potentially win bipartisan support.

One of the clearest impediments to mass killings, gun control advocates say, is a limit on the size of magazines permitted for civilian gun owners.

‘’I’ve never known a gun owner in Vermont who says, ‘No, I’ve got to have 30 rounds,’” Leahy said.

A measure limiting magazines has been drawing bipartisan backing since the school massacre in Newtown, Conn.

‘’I’m going to support the limitation on the size of the clips,” said Sen. Susan Collins, R-Maine. Such a bill, narrowly written, “might well pass,” she said.

Sen. Angus King of Maine, the newly elected Independent, has said that he supports universal background checks and limits on high-capacity ammunition magazines as well, but is waiting to see specific legislative language on all of the proposals, including an assault weapons ban.

The biggest problem may be marrying too many components in one bill. For example, a measure that Sen. Dianne Feinstein, D-Calif., is working on would reinstitute the ban on assault weapons but also limit magazines. Such a measure would probably enjoy the support of some big-city mayors and the like, but could be met coolly by other Democrats, particularly those from conservative states who are up for re-election in 2014.

Any legislation that managed to get through the Senate would have a tougher ride in the House.

‘’The necessity of having that legislation here instead of in the states is lost on me,” said Rep. Rob Woodall, R-Ga. “Folks in New York may have less of that grandfather-to-grandson exchange that we have in Georgia that takes place under the Christmas tree every year.”

Leahy said legislation would be shaped after he heard from a range of witnesses in his hearings. He said he had invited Wayne LaPierre, the chief executive of the National Rifle Association.

The NRA did not respond to an inquiry about whether LaPierre would attend.